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)

Let $W$ is a finite dimensional subspace of an inner product space $V$ and $y$ is any vector in $V$. The best approximation to $y$ from $W$ is then

• A) $Proj^{y}_{w}$
• B) $Proj^{w}_{y}$
• C)
• D)

The two vectors are said to be equivalent if:

• A) None of the above
• B) Both (a) and (b).
• C) Same length.
• D) Same direction

The matrix A^T(Transpose of A) x A is invertible if and only if the columns of A are _________.

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

The matrix [Math Processing Error]ATA${A^T}A$ is invertible If and only if the columns of A are ------

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

A m ×n matrix U has orthonormal columns if and only if [Math Processing Error]$${U^t}U = I$$

• A) False
• B) True
• C)
• D)

The only 0-dimensional subspace of {R^3} is

• A) Infinite space
• B) \left\{ {a,b} \right\}
• C) None of the above
• D) Zero space