Linear Algebrea Homework Help?

Question

I think have these answers right, but I was wondering if someone could help me make sure...

A is a 3x3 matrix. The null space of A = {The Zero Vector}
Select any of the following that is necessarily TRUE.
There may be more than one right answer.

a. the dimension of the null space = 1
b. the range of A = R^3
c. the columns of A are a basis for R^3
d. A defines a one to one mapping

I'm pretty sure D is true because when the null space is only equal the zero vector the function is one to one, and A must be true because there is only one vector in the null set. I'm pretty sure B and C are not necessarily true, but I'm not positive.

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T is a linear mapping from R^3 into R^3.
The range of T is a plane through the origin.
Which of the following describes the null space of T?

a. R^3
b. a plane
c. a line
d. a point

I think it's B because I remember reading that the range is parallel to the domain's null set, but, again, i'm not sure.

Answer 1

I haven't taken a linear algebra course for over 30 years, but I don't think that {0} is defined as having dimension 1. However, I have trouble arguing that it doesn't ...

Anyhow, a 1-1 linear mapping from R3 to itself will have range R3. So d being true tells you that b is true as well.

If there were a linear dependency among the columns, the coefficients would be a non-trivial member of the null space. So c is true as well.

And if the range of a mapping is a plane, the null space will be a line perpendicular (normal) to the plane.

Answer 2

First the setup. enable X and Y be the $ invested in the two bonds. So all of us know: X + Y = 4000 (volume invested) .04X + .06Y = 188 (earnings from 2 investments) Now resolve and get X = $ 2600 and Y = $ 1400 there are a number of the thank you to resolve. an ordinary way if to multiply the best equation via 0.04 after which subtract the result from the seocnd equation. .04 X + .04 Y = .04(4000) = one hundred sixty subtract this from the 2d equation: .04X + .06Y = 188 and get ... ------------------------------------ 0 + 0.02 Y = 188 –one hundred sixty = 28 So Y = 28/0.02 = 1400 X = 4000 – Y = 4000 – 1400 = 2600 wish that's sparkling. maximum human beings decide to sparkling decimals, yet whilst they're this easy, i like to apply that reality.