If we divide a non-zero vector by its length we get a

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

If $u$ and $v$ are non zero vectors in either $R^2$ or $R^3$ then by the law of cosines $||u-vII^2$=---------

• A) $||u-v||^2=||u||^2+||v||^2-2||u||||v||cos \theta$
• B) $||u-v||^2=||u||^2+||v||^2+2||u||||v||cos \theta$
• C)
• D)

Let u, v and w be vectors in ${R^n}$, then

• A) (u+v).w=u.w+v.w-2u.w
• B) None of the above
• C) (u+v).w=u.w-v.w
• D) (u+v).w=u.w+v.w

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

• A) 2
• B) $\sqrt{6}$
• C) 0
• D) $\sqrt {7}$

An n x n matrix A is _________ if and only if A has n linearly independent vectors.

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

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

• A) TRUE
• B) FALSE
• C)
• D)

Suppose that \mathop x\limits^{^} satisfies ${A^T}A\mathop x\limits^{^} = {A^T}b$, then we can say that $b-A\mathop x\limits^{^}$ is ------------ to the rows of ${A^T}$ and hence is orthogonal to the -------------- of $A$.

• A) Orthogonal, rows
• B) Orthogonal, column
• C) Parallel, columns
• D) Parallel, rows

Any orthogonal subsets of vector in an inner product space is linearly independent

• A) False
• B) True
• C)
• D)

Let W be a subspace of $${R^n}$$ y is 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 $$\parallel y - \widehat y\parallel \angle \parallel y - v\parallel$$ The vector $$\widehat y$$ is called the best approximation to y by elements of W.

• A) False
• B) True
• C)
• D)

If two rows are _________, they are linearly independent.

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

Which of the following set of vectors is NOT an orthogonal set?

• A) (3, 0, 0) ;( 0, 1, 0)
• B) (1, 1, 1) ;( 1, 0,-1)
• C) (2, 3) ;( -6, 4)
• D) (1, 2, 0) ;( 1, 0, 3)

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

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

The zero vector is orthogonal to every vector in $R^n$

• A) True
• B) False
• C)
• D)

Let W be a subspace of ${R^n}$ and $\{ {u_1},{u_2},...,{u_p}\}$ is any orthogonal basis of W, then $\mathop y\limits^\^ = {c_1}{u_1} + {c_2}{u_2} + ... + {c_n}{u_n}$ Where

• A) ${c_j} = \frac{{y.{u_j}}}{{{u_j}.{u_j}}}$
• B) ${c_j} = \frac{{y.{u_j}}}{{u_j}}$
• C) None of these
• D) ${c_j} = \frac{{y.{u_j}}}{y}$

Let $$s = \{ {u_1} + {u_2} + ......{u_p}\}$$ be the set of non-zero vectors in $${R^n}$$ is said to be an orthogonal set if all vectors in S are mutually orthogonal. That is O∉ S and $${u_i}.{u_j} = 0\,$$ ∀ i≠ j, i,j=1,2,……..p.

• A) True
• B) False
• C)
• D)

In least square problem, the vector $Ax$ will necessarily be in the ---------------, no matter what $‘x’$ we choose.

• A) Row space Row $A$
• B) Column space $ColA$
• C)
• D)

A m ×n matrix U has orthonormal columns if and only if $${U^t}U = I$$

• A) True
• B) False
• C)
• D)

Let W be a subspace of ${R^n}$ and $\{ {u_1},{u_2},...,{u_p}\}$ is an orthonormal basis, then

• A) $\Pr o{j_w}y = (y.{u_1}){u_1} + (y.{u_2}){u_2} + ... + (y.{u_p}){u_p}$
• B) $\Pr o{j_w}y = (y.y){u_1} + (y.y){u_2} + ... + (y.y){u_p}$
• C)
• D)

If u=(1, k, 0) and v=(1, 1, 0) are orthogonal vectors then which of the following is the value of k?

• A) k = 0
• B) k = -1
• C) k = 2
• D) k = 1

For any vectors u and v, the length of vector u – v will be ___________.

• A) || u – v ||
• B) | u – v |
• C) u . v
• D) || u . v ||