If there is a vector $v=(2, 1, 0)$ then $||v||$ is ------

• A) 2
• B) 0
• C) 3
• D) $\sqrt{5}$

If two rows are orthogonal, they are __________.

• A) linearly independent
• B) linearly dependent
• C)
• D)

How many subspaces does $R^2$ have?

• A) $0, R×0,0×R,R^2$
• B) None of these
• C) Infinitely many
• D) $0$ and $R^2$

Two vectors are linearly independent if at least one of the vector is a multiple of the other.

• A) FALSE
• B) TRUE
• C)
• D)

Let W be a subspace of $R^n$, y be any vector in $R^n$ and $\widehat y$ the orthogonal projection of $y$ onto $W$. Then $\widehat y$ is the closest point in $W$ to $y$, in the sense that

• A) $||y-\widehat y ||>||y-v||$
• B) $||y-\widehat y||<||y-v||$
• C)
• D)

Let V be an vector space, and let W be a subset of V. What does it mean when we say that W is closed under addition?

• A) $W(x+y)=Wx+Wy$ for every two vectors $x$ and $y$
• B) Whenever $x$ and $y$ are in $V$, then x+y is in $V$
• C)
• D)

$${\text{The }}\_\_\_\_\_ ~vector~ is ~orthogonal~ to ~every ~vector ~in~ {R^n}$$

• A) unit
• B) none of these
• C) zero
• D) normalized

An n x n matrix A is invertible if and only if ___________ is not an eigen value of A.

• A) 1
• B) 0
• C) 2
• D) -1

A ________ matrix with orthogonal columns is an orthogonal matrix.

• A) singular
• B) diagonal
• C) rectangular
• D) square

Which of the following set of vectors is an orthogonal set?

• A) (0, 1, 1) ;( 1, 0,-1)
• B) (1, 0, 1) ;( -1, 0, -1)
• C) (1, 1, 1) ;( 1, 0,-1)
• D) (1, 1, 0) ;( 1, 0,-1)

A vector whose length is 1 is called -----------

• A) Identity Vector
• B) Unit Vector
• C)
• D)

Any set of three non-coplanar vectors {a,b,c} in ordinary space spans the -------- space

• A) ${R^3}$
• B) ${R^2}$
• C)
• D)

Elementary row operations on a matrix do not affect the -------- relations among the columns of the matrix.

• A) Linear dependence
• B) Linear independence
• C)
• D)

If a matrix U has orthonormal columns, then UUT (transpose of U) = ________.

• A) 0
• B) positive
• C) I
• D) negative

If $u$ and $v$ are vectors in $R^n$, then we regard $u$ and $v$ as $n \times 1$ matrices. The matrix product $u^{t}v$ is a ----------- matrix.

• A) $1 \times n$
• B) $1 \times 1$
• C) $n \times 1$
• D) $n \times n$

A matrix with orthogonal columns is an _________ matrix.

• A) orthonormal
• B) identity
• C) diagonal
• D) orthogonal

The product of upper triangular matrices is

• A) Diagonal matrix.
• B) None of the above
• C) Upper triangular matrix
• D) Lower triangular matrix.

A square matrix with orthogonal columns is ________ matrix.

• A) an nonivertible
• B) an orthonormal
• C) an invertible
• D) an orthogonal

If B=$$\{ {v_1},{v_2},{v_3}\}$$ is an orthogonal set of vectors with respect to an inner product on a vector space V, then the set B

• A) Spans the vector space v
• B) Linearly dependent.
• C) Is an orthonormal basis for v
• D) Is linearly independent.

If x is _______ to both u and v, then x must be orthogonal to u – v.

• A) orthogonal
• B) orthonormal
• C)
• D)