If three vectors are on the z-x plane and the fourth vector is not in this plane, the vectors would be linearly independent, no?

Ishaan Booker
2022-07-15
Answered

Are four non zero vectors are always linearly dependant

If three vectors are on the z-x plane and the fourth vector is not in this plane, the vectors would be linearly independent, no?

If three vectors are on the z-x plane and the fourth vector is not in this plane, the vectors would be linearly independent, no?

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asked 2021-05-29

Find a vector equation and parametric equations for the line segment that joins P to Q.

P(0, - 1, 1), Q(1/2, 1/3, 1/4)

P(0, - 1, 1), Q(1/2, 1/3, 1/4)

asked 2021-05-29

Which of the following expressions are meaningful? Which are meaningless? Explain.

a)$(a\cdot b)\cdot c$

$(a\cdot b)\cdot c$ has ? because it is the dot product of ?.

b)$(a\cdot b)c$

$(a\cdot b)c$ has ? because it is a scalar multiple of ?.

c)$|a|(b\cdot c)$

$|a|(b\cdot c)$ has ? because it is the product of ?.

d)$a\cdot (b+c)$

$a\cdot (b+c)$ has ? because it is the dot product of ?.

e)$a\cdot b+c$

$a\cdot b+c$ has ? because it is the sum of ?.

f)$|a|\cdot (b+c)$

$|a|\cdot (b+c)$ has ? because it is the dot product of ?.

a)

b)

c)

d)

e)

f)

asked 2021-02-11

Let F be a fixed 3x2 matrix, and let H be the set of all matrices A in $M}_{2\times 4$ with the property that FA = 0 (the zero matrix in ${M}_{3\times 4})$ . Determine if H is a subspace of $M}_{2\times 4$

asked 2021-05-17

Find the scalar and vector projections of b onto a.

$a=(4,7,-4),b=(3,-1,1)$

asked 2020-10-21

This is the quesetion. Suppose that a does not equal 0.

a. if

b. if

c. if

Either prove the assertion is true in general or show that it is false for a concret choice of vectors a, b, c

asked 2021-12-14

Projection of u onto v and v onto u

Given the vector$u=<-2,6,4>$ and a vector v such that the vector projection of u onto v is $<2,4,4>$ , and the vector projection of v onto u is $<-8,24,16>$ . What is the vector v?

Let$\overrightarrow{v}=<a,b,c>$

Projection of$\overrightarrow{v}\text{}on\to \text{}\overrightarrow{u}$ is given by:

$pro{j}_{u}v=\frac{\overrightarrow{u}.\overrightarrow{v}}{{\left|u\right|}^{2}}\overrightarrow{u}=\frac{\overrightarrow{u}.\overrightarrow{v}}{{(-2)}^{2}+{6}^{2}+{4}^{2}}<-2,6,4>$

$<-8,24,16\ge \frac{\overrightarrow{u}.\overrightarrow{v}}{{(-2)}^{2}+{6}^{2}+{4}^{2}}<-2,6,4>$

$\Rightarrow 4<-2,6,4\ge \frac{\overrightarrow{u}.\overrightarrow{v}}{4+36+16}<-2,6,4>$

$\Rightarrow <-2,6,4\ge \frac{\overrightarrow{u}.\overrightarrow{v}}{224}<-2,6,4>$

On comparing$\frac{\overrightarrow{u}.\overrightarrow{v}}{224}=1$

$\Rightarrow \overrightarrow{u}.\overrightarrow{v}=224$

$pro{j}_{v}u=\frac{\overrightarrow{u}.\overrightarrow{v}}{{\left|v\right|}^{2}}\overrightarrow{v}=\frac{224}{{\left|v\right|}^{2}}<a,b,c>$

$\Rightarrow <2,4,4\ge \frac{224}{{\left|v\right|}^{2}}<a,b,c>$

Dividing both sides by 2 we get:

$\Rightarrow <1,2,2\ge \frac{112}{{\left|v\right|}^{2}}<a,b,c>$

Given the vector

Let

Projection of

On comparing

Dividing both sides by 2 we get:

asked 2022-09-16

In my syllabus we have the alternative definition of the condition of a matrix:

$\kappa (A)=\frac{{\text{max}}_{\Vert \overrightarrow{y}\Vert =1}\Vert A\overrightarrow{y}\Vert}{{\text{min}}_{\Vert \overrightarrow{y}\Vert =1}\Vert A\overrightarrow{y}\Vert}$

In it, it also says that by definition of the condition of a matrix it follows that $\kappa ({A}^{-1})=\kappa (A)$. So there is no explanation for this. Therefore, my question is: Why is $\kappa ({A}^{-1})=\kappa (A)$

$\kappa (A)=\frac{{\text{max}}_{\Vert \overrightarrow{y}\Vert =1}\Vert A\overrightarrow{y}\Vert}{{\text{min}}_{\Vert \overrightarrow{y}\Vert =1}\Vert A\overrightarrow{y}\Vert}$

In it, it also says that by definition of the condition of a matrix it follows that $\kappa ({A}^{-1})=\kappa (A)$. So there is no explanation for this. Therefore, my question is: Why is $\kappa ({A}^{-1})=\kappa (A)$