Projection in Linear Algebra: Definition, Properties, and Examples

Question 1

Which of the following statements accurately defines the projection of vector a onto subspace S?

Accepted Answer

A vector in subspace S that is parallel to vector a

Answer

A vector in subspace S that has the same magnitude as vector a

Answer

A vector that is perpendicular to subspace S and has the same magnitude as vector a

Answer

A vector that is the difference between vector a and its orthogonal projection onto subspace S

Question 2

Which of the examples below illustrates the concept of projection?

Accepted Answer

The shadow of an object on a screen

Answer

The movement of an object along a curved path

Answer

The trajectory of a projectile in the absence of air resistance

Answer

The motion of an electron around an atomic nucleus

Question 3

Which of the following statements about the projection of a vector onto a subspace is incorrect?

Accepted Answer

The projection is always shorter than or equal to the original vector.

Answer

The projection is unique.

Answer

The projection is perpendicular to the subspace.

Answer

The projection lies in the subspace.

Question 4

What is the dimension of the projection of a vector onto a subspace of dimension k?

Accepted Answer

k

Answer

The same as the dimension of the original vector

Answer

The sum of the dimensions of the subspace and the original vector

Answer

The difference between the dimensions of the subspace and the original vector

Question 5

What is the defining characteristic of the projection of a vector onto a subspace?

Accepted Answer

The projection is the vector in the subspace closest to the original vector.

Answer

The projection is the only vector in the subspace perpendicular to the original vector.

Answer

The projection is the vector in the subspace parallel to the original vector.

Question 6

In an inner product space, the projection of a vector v onto a subspace W is the vector:

Accepted Answer

that minimizes the distance between v and W

Answer

that is orthogonal to W

Answer

that is in the same direction as v

Answer

that is parallel to W

Question 7

What is the dimension of the orthogonal complement of a subspace W of an n-dimensional inner product space?

Accepted Answer

n - dim(W)

Answer

dim(W)

Answer

n + dim(W)

Answer

dim(W - n)

Question 8

True or False: The projection of a vector onto a subspace is unique.

Accepted Answer

True

Answer

False

Question 9

What is the geometric interpretation of projection?

Accepted Answer

It is the closest point on a subspace to a given vector.

Answer

It is the vector that is perpendicular to the subspace.

Answer

It is the vector that is parallel to the subspace.

Answer

It is the vector that is in the same direction as the given vector.

Question 10

In an inner product space, the orthogonal complement of a subspace W is the set of all vectors that are:

Accepted Answer

Perpendicular to every vector in W

Answer

Parallel to every vector in W

Answer

Linearly dependent of every vector in W

Answer

Spanned by every vector in W

Question 11

The projection of a vector u onto a subspace W is the vector that:

Accepted Answer

Minimizes the distance between u and W

Answer

Maximizes the distance between u and W

Answer

Is orthogonal to every vector not in W

Answer

Is parallel to W

Question 12

How can you find the projection of a vector u onto a subspace W spanned by a set of vectors {v1, v2, ... , vn}?

Accepted Answer

Use the formula proj_W(u) = *v1 + *v2 + ... + *vn

Answer

Compute the dot product of u with each vector in {v1, v2, ... , vn} and then sum the results

Answer

Use the Gram-Schmidt process to orthogonalize {v1, v2, ... , vn} and then project u onto each of the orthogonal vectors

Answer

Find a vector w in W that is closest to u and then take the projection of u onto w

Question 13

What is the relationship between the projection of a vector u onto a subspace W and the orthogonal component of u with respect to W?

Accepted Answer

The projection of u onto W is orthogonal to the orthogonal component of u with respect to W

Answer

The projection of u onto W is equal to the orthogonal component of u with respect to W

Answer

The projection of u onto W is the negative of the orthogonal component of u with respect to W

Answer

There is no relationship between the projection of u onto W and the orthogonal component of u with respect to W

Question 14

What is the dimension of the orthogonal complement of a subspace of dimension k in an n-dimensional inner product space?

Accepted Answer

n-k

Answer

k

Answer

n

Answer

k + n

Question 15

Which of the following is the projection of vector **u** onto vector **v** in an inner product space?

Accepted Answer

((**u** . **v**) / ||**v**||^2) **v**

Answer

u + v

Answer

u - v

Answer

((**v** . **u**) / ||**u**||^2) **u**

Question 16

If the inner product of vectors **u** and **v** is 0, then the projection of **u** onto **v** is:

Accepted Answer

0

Answer

u

Answer

v

Answer

u + v

Question 17

What is the dimension of the subspace spanned by the projection of vector **u** onto vector **v**?

Accepted Answer

1

Answer

2

Answer

3

Answer

0

Question 18

Which of the following operations is not the same as projection?

Accepted Answer

Eigenvalue decomposition

Answer

Orthogonal decomposition

Answer

Gram-Schmidt orthogonalization

Answer

Singular value decomposition

Question 19

Consider the vector space (R^3) with the standard inner product. What is the projection of the vector ((1, 2, 3)) onto the subspace spanned by the vector ((1, 1, 1))?

Accepted Answer

((1/3, 1/3, 1/3))

Answer

((2/3, 2/3, 2/3))

Answer

((1, 1, 1))

Answer

((0, 0, 0))

Question 20

Which of the following matrices (M) is a projection matrix?

Accepted Answer

(M = egin{bmatrix}1 & 0 \ 0 & 1end{bmatrix})

Answer

(M = egin{bmatrix}1 & 2 \ 3 & 4end{bmatrix})

Answer

(M = egin{bmatrix}1 & 0 \ 0 & -1end{bmatrix})

Answer

(M = egin{bmatrix}0 & 1 \ 1 & 0end{bmatrix})

Question 21

The projection of a vector onto a zero subspace is:

Accepted Answer

The zero vector

Answer

The original vector

Answer

A vector of the same direction

Answer

A vector of the same magnitude

Question 22

What is the orthogonal complement of the span of a set of vectors v1, v2, ..., vn in an inner product space?

Accepted Answer

The subspace of all vectors that are orthogonal to every vector in the span

Answer

The subspace of all vectors that are parallel to every vector in the span

Answer

The subspace of all vectors that are linear combinations of v1, v2, ..., vn

Answer

The subspace of all vectors that are not in the span

Question 23

What is one of the properties of the projection of a vector v onto a vector u in an inner product space?

Accepted Answer

The projection of v onto u is parallel to u

Answer

The projection of v onto u is always shorter than v

Answer

The projection of v onto u has the same direction as v

Answer

The projection of v onto u is unique

Question 24

What is the formula to calculate the projection of a vector v onto a vector u in an inner product space?

Accepted Answer

proj_u(v) = (v dot u) / (u dot u) * u

Answer

proj_u(v) = (u dot v) / (u dot u) * v

Answer

proj_u(v) = (v cross u) / (u cross u) * u

Answer

proj_u(v) = (v dot u) / (v dot v) * v

Question 25

What is an application of projection in linear algebra?

Accepted Answer

Finding the closest point to a given point on a line

Answer

Solving systems of linear equations

Answer

Finding the eigenvectors of a matrix

Answer

Calculating the covariance of two random variables

Question 26

What is the rank of the projection matrix onto a subspace of dimension k?

Accepted Answer

k

Answer

k-1

Answer

k+1

Answer

n-k

Question 27

Which of the following operators in linear algebra is a projection operator?

Accepted Answer

An operator that is idempotent

Answer

The identity operator

Answer

The zero operator

Answer

An operator that is orthogonal

Question 28

What is the dimension of the image of a projection?

Accepted Answer

The same as the dimension of the subspace it projects onto

Answer

The same as the dimension of the vector space it operates on

Answer

The same as the rank of the projection operator

Answer

The same as the nullity of the projection operator

Question 29

In an inner product space, the projection of a vector u onto a subspace W is:

Accepted Answer

The vector in W that is closest to u.

Answer

The vector in W that is parallel to u.

Answer

The vector in W that has the same length as u.

Answer

The vector in W that is perpendicular to u.

Question 30

Which of the following best defines projection in the context of linear algebra?

Accepted Answer

The vector in the subspace that is closest to the given vector.

Answer

The vector in the subspace that is orthogonal to the given vector.

Answer

The vector in the subspace that has the same length as the given vector.

Question 31

How do we typically find the projection of a vector onto a subspace in linear algebra?

Accepted Answer

By finding the vector in the subspace that minimizes the distance to the given vector.

Answer

By finding the vector in the subspace that has the same length as the given vector.

Answer

By finding the vector in the subspace that is parallel to the given vector.

Question 32

Which of the following is equivalent to the projection of vector (v) onto subspace (W)?

Accepted Answer

The vector in (W) that is closest to (v).

Answer

The vector in (W) that is parallel to (v).

Answer

The vector in (W) that has the same length as (v).

Answer

The vector in (W) that is orthogonal to (v).

Question 33

Which of the following properties holds true about the projection of vector (v) onto subspace (W)?

Accepted Answer

The projection of (v) onto (W) always lies within the subspace (W).

Answer

The projection of (v) onto (W) is always orthogonal to (W).

Answer

The projection of (v) onto (W) is always parallel to (v).

Answer

The projection of (v) onto (W) always has the same length as (v).

Question 34

Consider an inner product space. Which of the following formulas correctly expresses the projection of vector (v) onto subspace (W)?

Accepted Answer

P = ((v, w) / (w, w)) * w

Answer

P = (v, w) / w

Answer

P = (v, w) * w

Answer

P = v / w

Question 35

In linear algebra, subspaces are commonly found in various mathematical contexts. Which of the following is an example of a subspace within the Euclidean space R³?

Accepted Answer

The plane defined by the equation x + y + z = 0.

Answer

The circle defined by the equation x² + y² = 1.

Answer

The sphere represented by the equation x² + y² + z² = 1.

Answer

The curve described as y = 2x.

Question 36

Among the applications of projection in the field of linear algebra, which of the following is a notable example?

Accepted Answer

The least squares method, employed in approximating and fitting data.

Answer

Gaussian elimination for solving systems of linear equations.

Answer

Calculating the determinant of a matrix.

Question 37

In an inner product space, let u and v be orthogonal vectors. What is the projection of u onto v?

Accepted Answer

0

Answer

v

Answer

u

Answer

||u||

Question 38

The projection matrix onto a subspace W is given by the formula:

Accepted Answer

P = W * (W^T W)^-1 * W^T

Answer

P = W^T * (W W^T)^-1 * W

Answer

P = W^T * (W^T W)^-1 * W^-1

Answer

P = (W^T W)^-1 * W W^T

Question 39

The dimension of the projection of a vector v onto a subspace W is:

Accepted Answer

equal to the dimension of W

Answer

less than the dimension of W

Answer

equal to the dimension of v

Answer

greater than the dimension of W

Question 40

Which of the following is equivalent to the orthogonal projection of a vector v onto a subspace W?

Accepted Answer

The vector in W closest to v in terms of distance.

Answer

The vector in W that has the same length as v.

Answer

The vector in W that is parallel to v.

Question 41

Which of the following matrices is NOT a projection matrix?

Accepted Answer

A matrix that is not symmetric

Answer

A matrix that is idempotent

Answer

A matrix that has eigenvalues of 0 and 1

Answer

A matrix that has trace 1

Question 42

Let  "u" and  "v" be vectors in an inner product space. Which of the following is **not** a property of the projection of  "u" onto  "v"?

Accepted Answer

The projection of "u" onto "v" is always orthogonal to "u" itself.

Answer

The projection of "u" onto "v" is always in the direction of "v".

Answer

The projection of "u" onto "v" is the closest point on the line spanned by "v" to "u".

Question 43

In a vector space with an inner product, the projection of a vector "u" onto a subspace "W" is:

Accepted Answer

The closest point in "W" to "u".

Answer

The sum of all vectors in "W" that are orthogonal to "u".

Answer

The vector in "W" that is orthogonal to "u".

Answer

The vector in "W" that has the same direction as "u".

Question 44

Which of the following is **not** a direct application of projections in linear algebra?

Accepted Answer

Finding the eigenvalues of a matrix

Answer

Finding the orthogonal complement of a subspace

Answer

Solving least squares problems

Question 45

The projection of a vector onto itself is:

Accepted Answer

The vector itself

Answer

Half the length of the original vector

Answer

The zero vector

Answer

A vector orthogonal to the original vector

Question 46

Let "u" and "v" be orthogonal vectors. What is the projection of "u" onto "v"?

Accepted Answer

The zero vector

Answer

The vector "u"

Answer

The vector "v"

Answer

A vector orthogonal to both "u" and "v"

Question 47

Let "W" be a subspace of an inner product space. The orthogonal complement of "W" is:

Accepted Answer

The set of all vectors orthogonal to every vector in "W"

Answer

The set of all vectors in "W"

Answer

The set of all vectors that are not in "W"

Question 48

The projection of a vector onto a line can be interpreted as:

Accepted Answer

The shadow of the vector cast on the line.

Answer

The reflection of the vector across the line.

Answer

The midpoint between the vector and the origin.

Answer

The angle between the vector and the line.

Question 49

Given a vector "u" and a subspace "W", the projection of "u" onto "W" can be used to decompose "u" into two components. These components are:

Accepted Answer

The projection of "u" onto "W" and the projection of "u" onto the orthogonal complement of "W".

Answer

Two vectors orthogonal to each other in "W".

Answer

The projection of "u" onto "W" and a vector parallel to "u".