Show that W, the set of all 3 times 3 upper triangular matrices, forms a subspace of all 3 times 3 matrices. What is the dimension of W? Find a basis for W.

Dottie Parra 2021-01-25 Answered
Show that W, the set of all 3×3 upper triangular matrices,
forms a subspace of all 3×3 matrices.
What is the dimension of W? Find a basis for W.
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Expert Answer

StrycharzT
Answered 2021-01-26 Author has 102 answers

To show that W, the set of all 3×3 upper triangular matrices,
forms asubspace of all 3×3 matrices.
Now we know that M, the set of all 3×3 matrices, forms a vector space.
Let A, B in W with A=[a1b1c10d1c100f1]B=[a2b2c20d2c200f2]
and k be a scalar. Now,
A+B=[a1b1c10d1c100f1]+[a2b2c20d2c200f2]=[a1+a2b1+b2c1+c20d1+d2c1+c200f1+f2]
Once again, kA=k[a1b1c10d1c100f1]
[ka1kb1kc10kd1kc100kf1] in W
Therefore, W is a subspace of M. Now let x is a typical element of W, with
X=[abc0dc00f]
can be written as A=aE11+bE12+cE13+dE22+cE23+fE33
where Eij is
3×3 time matrix with (i, j) element is 1 and rest are zore.
Now for any linear combination
x1E11+x2E12+x3E13+x4E22+x5E23+x6E33=03×3
imply x1=x2==x6=0
Therefore, E11,E12,E13,E22,E23,E33 forms a basis of W,
hence dim W=6.

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