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)

The set of least-square solutions of $Ax=b$ coincides with the nonempty set of solutions of the normal equation

• A) None of these
• B) $A^{T} A \widehat x =A^{T}b$
• C) $A^{T} A\widehat x =A^{T}$
• D) $A\widehat x=A^{T}$

Orthonormal set is a set of all vectors that are

• A) Mutually orthonormal and are of unit length
• B) Mutually orthonormal and of null length
• C)
• D)

The set of least square solutions of Ax=b coincides with the non-empty set of solutions of the normal equations

• A) ${A^T}\mathop x\limits^{-} A = b{A^T}$
• B) ${A^T}\mathop x\limits^{-} A = b{A^T}$
• C)
• D)

Let $$s = \{ {u_1} + {u_2} + ......{u_p}\}$$ be a basis for a subspace W of $${R^n}$$ , is also an orthogonal basis if S is an orthogonal set.

• A) False
• B) True
• C)
• D)

A square matrix with _______ columns is an orthogonal matrix.

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

If a _____ matrix has orthonormal columns, then it also has orthonormal rows.

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

Let B and C denote subsets of a vector space V.

• A) If B $$\subseteq$$C then span(C) = span(B).
• B) If B $$\subseteq$$C and C is dependent, then B is dependent.
• C) If B$$\subseteq$$ C and C spans V then B spans V.
• D) If B $$\subseteq$$C and C is independent then B is independent.

Two vectors are _______if at least one of the vector is a multiple of the other

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

The matrix ${A^T}A$ is invertible If and only if the columns of A are ------

• A) Linearly dependent
• B) None of the above
• C) Equal
• D) linearly independent

Let be a five-dimensional vector space, and let be a subset of consisting of five vectors. Then S

• A) Must be linearly independent, but cannot span V
• B) Must be linearly dependent, but may or may not span V
• C)
• D)

The matrix $A^{T}A$ is invertible iff the columns of A are-----------------------

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

Any finite dimensional inner product space possesses an orthonormal basis

• A) True
• B) False
• C)
• D)

For u and v vectors in $R^n$, thw distance between u and v written as $dist(u, v)$ is the length of the vector------

• A) $v-u$
• B) $u-v$
• C)
• D)

If A is an m x n matrix with linearly independent columns, then A can be factored as A = QR, where Q is an m x n matrix whose columns form an orthonormal basis for Col A and R is an n x n upper triangular invertible matrix with positive entries on its diagonal

• A) True
• B) False
• C)
• D)