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Linear Algebra : Operations and Properties

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Example Questions

Example Question #19 : Range And Null Space Of A Matrix

$C(-\infty , \infty)$ , the set of all continuous real-valued functions defined on $(-\infty, \infty)$ , is a vector space under the usual rules of addition and scalar multiplication.

Let be the set of all functions of the form

$f(x) = \left\{\begin{matrix} 2nx+2n , & x \le - 1 \\ 0, & -1 < x < 1\\ -nx+n & x \ge 1 \end{matrix}\right.$

for some real

True or false: is a subspace of $C(-\infty , \infty)$ .

Possible Answers:

True

False

Correct answer:

True

Explanation:

A set is a subspace of a vector space if and only if two conditions hold, both of which are tested here.

The first condition is closure under addition - that is:

If $a , b \in S$ , then $a+ b \in S$

Let $f , g \in S$ as defined. Then for some $n, m \in \mathbb{R}$ ,

$f(x) = \left\{\begin{matrix} 2nx+2n , & x \le - 1 \\ 0, & -1 < x < 1\\ -nx+n & x \ge 1 \end{matrix}\right.$

and

$g(x) = \left\{\begin{matrix} 2mx+2m, & x \le - 1 \\ 0, & -1 < x < 1\\ -mx+m & x \ge 1 \end{matrix}\right.$

Then

$(f+g)(x) = \left\{\begin{matrix}( 2nx+2n)+ ( 2mx+2m) , & x \le - 1 \\ 0+0 , & -1 < x < 1\\ (-nx+n) + (-mx+m) & x \ge 1 \end{matrix}\right.$

$(f+g)(x) = \left\{\begin{matrix}( 2nx+2mx )+ ( 2n +2m) , & x \le - 1 \\ 0+0 , & -1 < x < 1\\ (-nx+(-mx)) + (n+m) & x \ge 1 \end{matrix}\right.$

$(f+g)(x) = \left\{\begin{matrix} 2(n + m)x+ 2 ( n +m) , & x \le - 1 \\ 0 , & -1 < x < 1\\ -(n +m)x + (n+m) & x \ge 1 \end{matrix}\right.$

$f +g \in S$ . The first condition is met.

The second condition is closure under scalar multiplication - that is:

If $a \in S$ and is a scalar, then $ca \in S$

Let $h \in S$ as defined. Then for some $p \in \mathbb{R}$ ,

$h(x) = \left\{\begin{matrix} 2px+2p , & x \le - 1 \\ 0, & -1 < x < 1\\ -px+p & x \ge 1 \end{matrix}\right.$

For any scalar ,

$c h(x) = \left\{\begin{matrix} c (2px+2p) , & x \le - 1 \\ c \cdot 0, & -1 < x < 1\\ c(-px+p) & x \ge 1 \end{matrix}\right.$

$(c h)(x) = \left\{\begin{matrix} 2(cp)x+2 (c p) , & x \le - 1 \\ 0, & -1 < x < 1\\ -(cp)x+cp & x \ge 1 \end{matrix}\right.$

$c h \in S$ . The second condition is met.

, as defined, is a subspace.

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Example Question #12 : Range And Null Space Of A Matrix

If is an $m \times n$ matrix, find

dim(row(A))+dim(col(A))+dim(null(A))+dim(null(A^T)).

Possible Answers:

|n-m|

n+m

2n+2m

Correct answer:

n+m

Explanation:

Since a basis for the row space and the column space of a matrix have the same, number of vectors then their dimensions are the same, say .

By the rank-nullity theorem, we have rank(A) +Null(A) = n , or same to say

dim(row(A))+dim(null(A)) = n .

r +dim(null(A)) =n .

Hence dim(null(A))=n-r .

Finally, applying the rank-nullity theorem to the transpose of , we have

rank(A^T) +Null(A^T) = m , or the same to say

dim(row(A^T)) +dim(null(A^T)) = m .

r + dim(null(A^T))= m (The row space dimension of is the same as its transpose.)

dim(null(A))=r-m .

Adding all four of our findings together gives us

r+r+n-r+m-r = n+m .

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Example Question #1 : The Determinant

Calculate the determinant of matrix A where

$A = \begin{bmatrix} 5&4 \\ 1&2 \\ 0&7 \end{bmatrix}$

Possible Answers:

Not Possible

-50

Correct answer:

Not Possible

Explanation:

The matrix must be square to calculate its determinant, therefore, it is not possible to calculate the determinant for this matrix.

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Example Question #2 : The Determinant

Calculate the determinant of matrix A where,

$A=\begin{bmatrix} 4&1 \\ 5& 3 \end{bmatrix}$

Possible Answers:

-7

Correct answer:

Explanation:

To calculate the determinant of a 2x2 matrix, we can use the equation $a_{^{11}}a_{^{22}}-a_{^{12}}a_{^{21}}$

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Example Question #3 : The Determinant

Calculate the determinant of matrix A where,

$A=\begin{bmatrix} 8&-8 \\ 51&12 \end{bmatrix}$

Possible Answers:

-315

504

-504

Correct answer:

504

Explanation:

To calculate the determinant of a 2x2 matrix, we can use the equation $a_{^{11}}a_{^{22}}-a_{^{12}}a_{^{21}}$

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Example Question #4 : The Determinant

Calculate the determinant of matrix A where,

$A=\begin{bmatrix} 8&6 \\ 0&2 \end{bmatrix}$

Possible Answers:

-15

Correct answer:

Explanation:

To calculate the determinant of a 2x2 matrix, we can use the equation $a_{^{11}}a_{^{22}}-a_{^{12}}a_{^{21}}$

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Example Question #5 : The Determinant

Calculate the determinant of matrix A where,

$A=\begin{bmatrix} 2&5 &6 \\ 2&1 &0 \\ 5&3 &4 \end{bmatrix}$

Possible Answers:

-24

-26

Correct answer:

-26

Explanation:

Calculating the determinant of a 3x3 matrix is more difficult than a 2x2 matrix. To calculate the determinant of a 3x3 matrix, we use the following $\left | A\right |=a_{11}(a_{22}a_{33}-a_{23}a_{32})-a_{12}(a_{21}a_{33}-a_{23}a_{31})+a_{13}(a_{21}a_{32}-a_{22}a_{31})$

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Example Question #6 : The Determinant

Calculate the determinant of matrix A where,

$A=\begin{bmatrix} -5&7 &10 \\ -20&15 &2 \\ 0& 3& 8 \end{bmatrix}$

Possible Answers:

-49

-50

Correct answer:

-50

Explanation:

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Example Question #7 : The Determinant

Calculate the determinant of $\begin{bmatrix} 5&-2 \\ 1&3 \end{bmatrix}$ .

Possible Answers:

Correct answer:

Explanation:

By definition,

$\begin{vmatrix} a&b \\ c&d \end{vmatrix}= a(d)-b(c)$ ,

therefore,

$\begin{vmatrix} 5&-2 \\ 1&3 \end{vmatrix}= 5(3)-1(-2)= 17$ .

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Example Question #1 : The Determinant

Calculate the determinant of $\begin{bmatrix} 1&3 &1 \\ 0& 2&0 \\ 4&8 &2 \end{bmatrix}$

Possible Answers:

Correct answer:

Explanation:

For simplicity, we will find the determinant by expanding along the second row. Consider the following:

$\begin{vmatrix} 1&3 &1 \\ 0& 2&0 \\ 4&8 &2 \end{vmatrix}= -0[3(2)-1(8)]+2[1(2)-1(4)]-0[1(8)-3(4)]= -4$

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Linear Algebra : Operations and Properties

Study concepts, example questions & explanations for Linear Algebra

All Linear Algebra Resources

Example Questions

Example Question #19 : Range And Null Space Of A Matrix

Example Question #12 : Range And Null Space Of A Matrix

Example Question #1 : The Determinant

Example Question #2 : The Determinant

Example Question #3 : The Determinant

Example Question #4 : The Determinant

Example Question #5 : The Determinant

Example Question #6 : The Determinant

Example Question #7 : The Determinant

Example Question #1 : The Determinant

All Linear Algebra Resources

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