The dimension of the zero vector space $\left\{ 0 \right\}$ is defined to be

• A) 0
• B) 1
• C) None of these
• D) $\infty$

If vector space is not spanned by a finite set of vectors, then V is said to be --------

• A) Infinite-dimensional
• B) Finite-dimensional
• C)
• D)

$$If~ A ~is~ an~ m \times n ~matrix ~then$$

• A) $$rank A = \dim Nul A$$
• B) $$rank A + \dim NulA = n$$
• C) $$rank A = \dim NulA + n$$
• D) $$rank A = n$$

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

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

If $B=/{b_{1}, b_{2}, …, b_{n}/}$ and $E$ is the standard basis $/{e_{1}, e_{2}, …, e_{n}/}$ in $R^{n}$, then

• A) $[b_{1}]_{E}=b_{1}$
• B) $[b_{1}]_{E}=0$
• C)
• D)

\lambda is an eigenvalue of a matrix A if and only if the equation $(A-\lambdaI)x=0$ has a ---------

• A) Non-trivial solution
• B) Trivial Solution
• C)
• D)

A coordinate system on a set consists of a ______________ mapping of the points in the set into R^n

• A) One-to-one
• B) Onto
• C)
• D)

If A is an m x n, matrix, then
Rank (A) = the number of leading variables in the solution of Ax = 0
Nullity (A) = the number of parameters in the general solution of Ax = 0

• A) False
• B) True
• C)
• D)

In $\dim \left( {{R^n}} \right) = n$, the standard basis has ------- vectors

• A) 0
• B) mn
• C) n
• D) n+1

The dominant or principal eigenvector of a matrix is an eigenvector corresponding to the eigenvalue of _________ magnitude (for real numbers, largest absolute value) of that matrix.

• A) smallest
• B) largest
• C) zero
• D) all of these

If 3 is an eigenvalue of $A$ and $x$ is a corresponding eigenvector, then what is the eigenvalue of $A^2$ ?

• A) 3
• B) 6
• C) 12
• D) 9

The set $B={1,x,x^2,…,x^n}$ forms a basis for the vector space $P_n$ of polynomial of degree

• A) None of these
• B) $\leq n$
• C) $n$
• D) $\geq n$

If a vector space $V$ has a basis $B={b_1, b_2, …, b_n}$ then any set in $V$, containing more than $n$ vectors, must be -----------

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

A and its transpose matrix have ________ eigenvalues.

• A) different
• B) same
• C) same but opposite in sign
• D) different and positive

The rank of the matrix (m × n) where m<n cannot be more than?

• A) n
• B) m
• C)
• D)

If A is an n x n matrix, then the following statements are equivalent EXCEPT.

• A) I – A is singular.
• B) det(A) = 0
• C) A has nontrivial fixed points.
• D) det(I – A) = 0

The set of unit vectors ${i,j,k}$form a basis of

• A) $R$
• B) $R^4$
• C) $R^2$
• D) $R^3$

The determinant of identity matrix is?

• A) 0
• B) None of the mentioned
• C) Depends on the matrix
• D) 1

If $\lambda$ is an eigenvalue of a matrix $A$ and $x$ is a corresponding eigenvector, and if $k$ is any positive integer, then ${\lambda ^k}$ is an eigenvalue of ______ and $x$ is a corresponding eigenvector.

• A) $A$
• B) $A^k$
• C) $\lambda$
• D) $\lambda ^k$

Every finite dimensional vector space contains a/an

• A) Infinite set
• B) Basis
• C) Empty set
• D) None of the above