If the columns of a matrix are ________ then the matrix is invertible.

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

Among the three matrices$$A = \left( {\begin{array}{*{20}{c}} 1&0 \\\ 0&0 \end{array}} \right)$$ $$B = \left( {\begin{array}{*{20}{c}} 1&0 \\\ 0&0 \end{array}} \right)$$ $$C = \left( {\begin{array}{*{20}{c}} 0&1 \\\ 1&0 \end{array}} \right)$$ the matrix (matrices) which is (are) diagonalizable over the matrix (matrices) which is (are) diagonalizable over F_2 is (are)_______________________ is (are)_______________________

• A) B
• B) A
• C) NON OF Above
• D) C

If A is m\cross n and b is in $R^n$, a least square solution of $Ax=b$ is an $\widehat x$ in $R^n$ such that

• A) $||b-A\widehat x||>=||b-Ax|| \forall x \in R^n$
• B) $||b-A\widehat x ||<=||b-Ax|| \forall x \in R^n$
• C)
• D)

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

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

If $\{u_{1}, u_{2}, …, u_{p}\}$ is an orthonormal basis for a subspace $W$ of $R^n$, then ...

• A) $Proj_{w}y=(y.u_{1})u_{p}+(y.u_{2})u_{p}+…+(y.u_{p})u_{p}$
• B) $Proj_{w}y=(y.u_{1})u_{1}+(y.u_{2})u_{2}+…+(y.u_{p})u_{p}$
• C)
• D)

The matrix A^T(Transpose of A) x A is ________ if and only if the columns of A are linearly independent.

• A) singular
• B) symmetric
• C) invertible
• D) scalar

Two vectors u and v are orthogonal to each other if _____________.

• A) u - v = 0
• B) u . v = 0
• C) u + v = 0
• D) u . v = 1

The process of creating the unit vector u from v sometime called--------

• A) Diagonalizing v
• B) Normalizing v
• C)
• D)

Let $A=QR$ be a $QR$ factorization of $A$, then the equation $Ax=b$ has a unique least-squares solution, given by

• A) $x^{^}=RQ^{T}b$
• B) $x^{^}=R^{-1}Q^{T}b$
• C)
• D)

The product of upper triangular matrices is:

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

Two vectors $u$ and $v$ in $R^n$ are orthogonal if ------

• A) $u.v \neq 0$
• B) $u.v=0$
• C)
• D)

If u=(2, 0, -1) and v=(0, -1, 0) is an orthogonal set then which of the following has a norm of 1?

• A) v.u
• B) u.v
• C) v.v
• D) u.u

If $$s = \{ {u_1} + {u_2} + ......{u_p}\}$$ is an orthogonal basis for a subspace W of $${R^n}$$ Then each y in W can be uniquely expressed as a linear combination of $${u_1} + {u_2} + ......{u_p}$$ That is$${c_1}{u_1} + {c_2}{u_2} + ......{c_p}{u_p}$$ Where$$\frac{{y.{u_j}}}{{{u_j}.{u_j}}}$$

• A) True
• B) False
• C)
• D)

Let U and W denote nonzero subspaces of a vector space V where no two of U, W and V are equal. If dim V = 3 then:

• A) If dim(U$$\cap$$ W) = 1 then U +W = V.
• B) dim(U$$\cap$$ W) = 2
• C) If dim(U) = 2, then U +W = V.
• D) dim(U $$\cap$$W) = 3.

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)