$$\begin{gathered} if~ A ~is~ a~ 4 \times 5~ matrix~ with~ a~ two ~- {\text{dimentinal}}~ null space, ~ \hfill \\ what~ is~ rank ~of ~A \hfill \\\ \end{gathered}$$

• A) 3
• B) 6
• C) 5
• D) 7

Rank of the matrix A =$$\left[ {\begin{array}{*{20}{c}} 0&0&0&0 \\\ 4&6&1&0 \\\ 1&0&0&0 \end{array}} \right]$$

• A) 3
• B) 2
• C) 0
• D) 1

Rank of the matrix A=$$\left[ \begin{gathered} 0\,\,\,0\,\,\,0\,\,\,0 \hfill \\ 4\,\,\,6\,\,\,\,1\,\,\,\,0\, \hfill \\ 1\,\,\,\,\,0\,\,\,\,0\,\,\,\,0 \hfill \\\ \end{gathered} \right]$$

• A) 3
• B) 1
• C) 2
• D) 0

Let $V$ be a five-dimensional vector space, and let $S$ be a subset of $V$ which spans $V$. Then $S$

• A) Must have infinitely many elements
• B) Must have at most five elements
• C) Must be a basis for $V$
• D) Must be linearly dependent

If A is any matrix, then \[Rank(A) = Rank({A^T})\]

• A) True
• B) false
• C)
• D)

The vectors spaces $F(-\infty, +\infty)$ and $C(-\infty, +\infty)$ are ----------

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

The set of all solutions of $(A-\lambdaI)x$ is just the ……………………. Of the matrix $A$.

• A) Topological space
• B) Metric space
• C) None of the above
• D) Null space

The act of changing A into P^-1 AP is called a similarity transformation

• A) False
• B) True
• C)
• D)

The eigenvalues of a 4 by 4 matrix [A] are given as 2, -3, 1, 3 and 7. The $|det(A)|$ then is

• A) 530
• B) cannot be determined
• C) 546
• D) 25

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

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

Rank of the matrix A =\[\left[ {\begin{array}{*{20}{c}} 0&0&0&0 \\\ 4&6&1&0 \\\ 1&0&0&0 \end{array}} \right]\]

• A) 0
• B) 3
• C) 2
• D) 1

The point of intersection of axes in three dimension coordinates system is called?

• A) Ordinate
• B) Parallel pair order
• C) Collective coordinate
• D) Collective intercept

An n × n real matrix A is invertible if and only if the span of the rows of A is R^n

• A) True
• B) False
• C)
• D)

If $\lambda$ is an eigenvector of $A$, then every nonzero vector $x$ such that $Ax=\lambda x$ is called an ----------------- of $A$ corresponding to -----------

• A) Eigenvalue, $A$
• B) Eigenvector, $\lambda$
• C) Eigenvalue, $\lambda$
• D) Eigenvector, $A$

Every square matrix A has at least one ----------.

• A) Fixed point
• B) Critical point
• C)
• D)

$Dim(M_{m\times n})$=-------

• A) 0
• B) $m-n$
• C) $m+n$
• D) $mn$*

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

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

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 $B=/{b_{1}, b_{2}, …, b_{n}/}$ 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)

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

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