Can someone solve this, i have tried for 2 days!!

Nov 2019
1
0
graz
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Unfortunately its on german and my english ins't that good so i've translated it over google translate:
Given are the 2 × 3 matrices A1 = (1 0 1 0 1 2), A2 = (1 0 1 3 1 0),A3 = (5 0 5 6 5 6).
(a) Are these matrices of R2 × 3 linearly dependent? If yes, represent A3 as a linear combination of the other two.
(b) Does the span of the matrices {A1, A2, A3} form a subspace of R2 × 3? If so, what is the dimension of this subspace?
(c) What is the dimension of the R-vector space R2 × 3? Enter a base.
(d) Which dimension has the R-vector space C2 × 3? Enter a base.


Hope that someone can help me.
 
Jun 2013
1,112
590
Lebanon
(a) Find scalars \(\displaystyle x\) and \(\displaystyle y\) such that \(\displaystyle x A_1+y A_2=A_3\). This says that the three matrices are linearly dependent

(b) any set of matrices in \(\displaystyle \mathbb{R}^{2\times 3}\) span a subspace of \(\displaystyle \mathbb{R}^{2\times 3}\)
It follows from (a) that the dimension of this subspace is \(\displaystyle 2\)