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

3 years ago

The population of a city has increased by 15% since it was last measured if the current population is 43700 what was the previou

maks197457 [2]

Answer:

37,145

Step-by-step explanation:

43,700×15%= 6,555

43,700-6,555= 37,145

the previous population was 37,145

8 0

2 years ago

How many distinct triangles can be formed for which m∠A = 75°, a = 2, and b = 3?

antiseptic1488 [7]

We can use the Sine Law:a / sin A = b / sin B2 / sin 75° = 3 / sin B2 / 0.966 = 3 / sin B ( after that we will cross multiply )
2 sin B = 3 · 0.9662 sin B = 2.898sin B = 2.898 : 2sin B = 1.499 > 1 ( it is not possible )Answer: A ) No triangles can be formed.

8 0

3 years ago