Solving Vector Spaces Tasks: Basis and Linear Transformations

[peroAlex]

Hello, everybody!
I would really appreciate if someone could help me understand how to solve the following two tasks. I am not sure whether my translation is correct, so if, by any chance, you know a more appropriate terminology, please let me know. I am not fluent in writing matrices here on the forum, therefore I've attached my attempt at a solution as a picture.

1. The problem statement, all variables, and given/known data
Task 1: Prove that vectors $v_1 = (1,1,1)$, $v_2 = (1,1,2)$ and $v_3 = (1,2,3)$ form basis of vector space $\mathbb{R}^3$ and develop vector $y = (6,9,14)$ according to such basis.
Task 2: Linear transformations maps vectors forming standard basis into vectors (1,4,-1), (2,-3,1) and (4,3,-2). How would such transformation affect vector $v = (1,-1,2)$? Also, which vector is mapped into $w = (6,11,-4)$?

Homework Equations

If you believe I should disclose something here, please let me know. Those are basic linear algebra tasks, so I believe there should be no problem (and yet I fail to understand it).

The Attempt at a Solution

Please, check the attached picture.

 

[jedishrfu]

Start with the definition of a basis. Can the vectors be parallel?

 

[peroAlex]

Presumably, you mean the first task. Parallelity means that they lie along the same line passing through the origin, or that their cross product is zero. Parallel vectors are scalar multiples of one another. Also, vectors in the first case are linearly independent (determinant of matrix $A$ isn't zero).

 

[cpt_carrot]

For part (ii) of task 1 you want to write $y = av_1 + bv_2 + cv_3$ and find $a$, $b$ and $c$. What happens if you e.g. take the inner product of this equation with $v_1$?

 

[Ray Vickson]

Hello, everybody!
I would really appreciate if someone could help me understand how to solve the following two tasks. I am not sure whether my translation is correct, so if, by any chance, you know a more appropriate terminology, please let me know. I am not fluent in writing matrices here on the forum, therefore I've attached my attempt at a solution as a picture.

1. The problem statement, all variables, and given/known data
Task 1: Prove that vectors $v_1 = (1,1,1)$, $v_2 = (1,1,2)$ and $v_3 = (1,2,3)$ form basis of vector space $\mathbb{R}^3$ and develop vector $y = (6,9,14)$ according to such basis.
Task 2: Linear transformations maps vectors forming standard basis into vectors (1,4,-1), (2,-3,1) and (4,3,-2). How would such transformation affect vector $v = (1,-1,2)$? Also, which vector is mapped into $w = (6,11,-4)$?

Homework Equations

If you believe I should disclose something here, please let me know. Those are basic linear algebra tasks, so I believe there should be no problem (and yet I fail to understand it).

The Attempt at a Solution

Please, check the attached picture.

Forget matrices for a while; just concentrate on
(1) definitions of bases;
(2) how to check if a set of vectors forms a basis; and
(3) how to express a particular vector in terms of a particlular basis.

So: how would you deal with topic (1) above? After that, how would you carry out (2)? (Hint: linear equations and their solution methods are essential links in the process.)

Matrices can be helpful in shortening some operations and in expressing some facts in a compact manner. However, you can do pretty much all of the basics of linear algebra without ever, once, using a matrix. Sometimes, beginners get hung up on trying to use matrices for everything, when all they really do is get in the way, and actually hinder understanding.

But: if you really insist on using a matrix, you can very easily type it out nicely here in LaTeX. This Forum makes available two "styles" for matrices, and which one you should use is a matter of personal taste.
$$A = \begin{bmatrix} a_{11}& a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix}$$
or
$$A = \pmatrix{ a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} }$$

To see how this was done, just right-click on the formula and choose the "display math as tex commands" option.

 

[jedishrfu]

Presumably, you mean the first task. Parallelity means that they lie along the same line passing through the origin, or that their cross product is zero. Parallel vectors are scalar multiples of one another. Also, vectors in the first case are linearly independent (determinant of matrix $A$ isn't zero).

But what is the definition of a basis?

https://en.wikipedia.org/wiki/Basis_(linear_algebra)

Can you construct the third vector by adding multiples of the other two?

 

[peroAlex]

OK, first of all, I would like to thank everyone for helping me on this question. I honestly couldn't wish for a better support, I truly appreciate your time and effort. thank you!

------------

I now understand that for task 1 (second part of it) I need to solve a system of equations that originates from suggestion from @cpt_carrot $$ y = a v_1 + b v_2 + c v_3 $$ This is essentially a linear system of equations ($a + b + c = 6$...) Solving this yields correct answer!
Bases allow one to represent vectors by a sequence of scalars called components. It's a finite set of vectors that are linearly independent. So every element can be expressed as a finite sum (linear combination) of basis vectors! That simplified understanding so much!

For task 2 I'm still struggling though. If a linear transformation maps vectors forming standard basis into a set of some other vectors, what type of linear equation would best portray that? Is it $$a i + b j + c k = v_1 + v_2 + v_3 $$

 

[Mark44]

Task 2: Linear transformations maps vectors forming standard basis into vectors (1,4,-1), (2,-3,1) and (4,3,-2). How would such transformation affect vector $v = (1,-1,2)$? Also, which vector is mapped into $w = (6,11,-4)$?

The first sentence says that T(1, 0, 0) = (1, 4, -1), T(0, 1, 0) = (2, -3, 1), and T(0, 0, 1) = (4, 3, -2). T(v) = T(1, -1, 2), which can be rewritten as a linear combination of the transformations of the three basis vectors.

For the second part, you want to find a vector u = (a, b, c) such that T(u) = (6, 11, -4).