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

• 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)

Let $W$ be a subspace of $R^n$, then each $y$ in $R^n$ can be written uniquely in the form

• A) none of the above
• B) $y=\widehat y +z$
• C) $y=\widehat y /z$
• D) $y=\widehat y -z$

If u=(3, 1, 1) and v=(-1, 2, 1) be two vectors then which of the following step is true to show that it is an orthogonal set?

• A) Norm of v=1+4+1 = 6
• B) u/v= (3+1+1)/( -1+2+1)= 5/2
• C) Norm of u=9+1+1 = 11
• D) u.v=-3+2+1 = 0

7x + 2y = 16
–21x – 6y = 24
The above system has a ___________ solution.

• A) consistent
• B) trivial
• C) inconsistent
• D) unique

The norm of $v$ is the non-negative scalar$||v||$ defined by

• A) $||v||=v_{1}^{2}+v_{2}^{2}+...v_{n}^{2}$
• B) $||v||=\sqrt{v_{1}^{2}+v_{2}^{2}+...v_{n}^{2}}$
• C)
• D)

Suppose x,y,z are some vectors in an inner product space (v,<,>) such that <x,y>=<x,z> for all x∈V, then y=z

• A) False
• B) True
• C)
• D)

Suppose that $A^{T}A$ is invertible then the equation $Ax=b$ has only one least-squares solution $\widehat x$ and it is given by

• A) $\widehat x =(A^{T}A)A^{T}b$
• B) $\widehat x =(A^{T}A)^(-1)A^{T}b$
• C)
• D)