A unit vector is obtained by ------

• A) $u={||v||}/{||v||}
• B) $u=v/{||v||}
• C) None of the above
• D) $u={||v||}/v

The equation $b=A\mathop x\limits^{^}+(b-A\mathop x\limits^{^}$ is a decomposition of $b$ into the sum of a vector in ------------ and a vector orthogonal to $ColA$.

• A) $RowA$
• B) $ColA$
• C)
• D)

If x is orthogonal to both u and v, then x must be _______ to u – v.

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

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

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

For two vectors to be orthonormal, the vectors are also said to be orthogonal. The reverse of the same

• A) Is not true
• B) Is true
• C)
• D)

The two vectors are said to be equivalent if

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

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) ${c_j} = \frac{{y.{u_j}}}{y}$