2/3x - 1/2 =-6/7 solve for x

Given A is an $m\times n$ matrix. Let $T :U\to V$ be the corresponding linear transformationover the field F and $\theta$ be identity vector in V. Now if $x\in Ker( T)\implies T(x)=\theta$ .

(1) The kernel of a linear transformation is a vector space : True.

Let $x,y\in Ker( T)$ , then,

$T(x+y)=T(x)+T(y)=\theta+\theta=\theta\impies x+y\in Ker( T)$

hence the kernel is closed under addition.

Let $\lambda\in F, x\in Ker( T)$ , then

$T(\lambda x)=\lambda T(x)=\lambda\times \theta=\theta$

$\lambda x\in Ker(T)$ and thus Ket(T) is closed under multiplication

Finally, fore all vectors $u\in U$ ,

$T(\theta)=T(\theta+(-\theta))=T(\theta)+T(-\theta)=T(\theta)-T(\theta)=\theta$

$\implies \theta\in Ker(T)$

Thus Ker(T) is a subspace.

(2) If the equation Ax=b is consistent, then Col(A) is $\mathbb R^m$ : False

if the equation Ax=b is consistent, then Col(A) must be consistent for all b.

(3) The null space of an mxn matrix is in $\mathbb R^m$

: False

The null space that is dimension of solution space of an m x n matrix is always in $\mathbb R^n$ .

(4) The column space of A is the range of the mapping $x\to Ax$

: True.

(5) Col(A) is the set of all vectors that can be written as Ax for some x. : True.

Here Ax will give a linear combination of column of A as a weights of x.

(6) The null space of A is the solution set of the equation Ax=0.

: True

5 0

2 years ago

Which of the ratio equivalent to 1:3 select all that apply

nekit [7.7K]

Answer:

4:12

6:18

8:24

Two ratios that have the same value are called equivalent ratios. To find an equivalent ratio, multiply or divide both quantities by the same number.

3 0

2 years ago

Question 5 Multiple Choice Worth 1 points) (04.05 MC) Determine if the ordered pair (-1 , -5) is a solution to the inequality y≤

Zielflug [23.3K]

Answer:

(-1, -5) satisfies the inequality and will be the solution to the inequality.

From the attached figure, it is clear that (-1, -5) is below the line.

Therefore, the correct option is:

''Yes, because 1-1, 5) is on the line''.

Step-by-step explanation:

Given the inequality

$y\le -\frac{3}{4}x-1$