Is Every Vector in Set W a Linear Combination of W1 and W2?

[hkus10]

1.) The set W of all 2x3 matrices of the form

a  b  c
a  0  0

where c = a + b, is a subspace of M23 (Matrics 23). Show that every vector in W is a linear combination of
W1 =

1  0  1
1  0  0

W2 =

0  1  1
0  0  0

Do I have to combine both W1 and W2 into one equation?

 

[Mark44]

1.) The set W of all 2x3 matrices of the form

a  b  c
a  0  0

where c = a + b, is a subspace of M23 (Matrics 23). Show that every vector in W is a linear combination of
W1 =

1  0  1
1  0  0

W2 =

0  1  1
0  0  0

Do I have to combine both W1 and W2 into one equation?

Yes. If A is any vector in this set, you need to show that there are constants c₁ and c₂ such that A = c₁W₁ + c₂W₂.

 

[hkus10]

can you give some more hints for solving this problem?

 

[hkus10]

What I get is

a  b  2c
a  0   0

which is not

a  b  c
a  0  0

 

[fzero]

You made some sort of mistake. What linear combination of $$W_1$$ and $$W_2$$ gave that?


Use Mark44's hint and compute

$$c_1 W_1 + c_2 W_2.$$

Then try to find $$c_1$$ and $$c_2$$ so that

$$c_1 W_1 + c_2 W_2 = \begin{pmatrix} a & b & a+b \\ a & 0 & 0\end{pmatrix}.$$

 

[hkus10]

You made some sort of mistake. What linear combination of $$W_1$$ and $$W_2$$ gave that?


Use Mark44's hint and compute

$$c_1 W_1 + c_2 W_2.$$

Then try to find $$c_1$$ and $$c_2$$ so that

$$c_1 W_1 + c_2 W_2 = \begin{pmatrix} a & b & a+b \\ a & 0 & 0\end{pmatrix}.$$

The answer is $$aW_1 + bW_2 = \begin{pmatrix} a & b & a+b \\ a & 0 & 0\end{pmatrix}.$$? a, b can be any real number?

 

[Mark44]

Solve this equation for c₁ and c₂.
$$c_1\begin{bmatrix} 1 & 0 & 1\\1 & 0 & 0\end{bmatrix} + c_2 \begin{bmatrix} 0 & 1 & 1\\0 & 0 & 0\end{bmatrix} = \begin{bmatrix} a& b & a + b\\a & 0 & 0\end{bmatrix}$$

For two matrices to be equal, their corresponding components have to be equal.