NEED HELP ASAP WILL GIVE BRAINLIEST

Mathematics

2 answers:

ollegr [7]3 years ago

6 0

Answer:

D. (272, 127)

Step-by-step explanation:

It's actually not A, I just took the test. The right answer is D.

Anuta_ua [19.1K]3 years ago

3 0

Answer:

<em>it's either A or B hope I helped a little</em>

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Show that the sets of upper-triangular and lower-triangular matrices are subspaces of F n,n . Call them U and L respectively. Wh

soldi70 [24.7K]

Answer:

dim L = dim U = $\frac{n(n+1)}{2}$

Step-by-step explanation:

We can do it only for the lower-triangular matrices, the case of the upper-triangular matrices is similar. We might caracterice nxn the lower-triangular matrices, as the nxn matrices $A=(a_{ij})$ such that the entry $a_{ij}=0$ if i<j.

Now let $A=(a_{ij})$ and $B=(b_{ij})$ be two lower triangular matrices, now if

$C=A+\lambda B$ for some $\lambda \in \mathbb{F}$

then the entry $c_{ij}$ of C is equal to

$c_{ij}=a_{ij}+\lambda b_{ij}$

Now, if i<j, it must hold that $a_{ij}=0 \quad \text{and} \quad b_{ij}=0$ . Therefore, if this is the case we must have that $c_{ij}=0$ and so we get that C is also a lower triangular matrix. This showa that L is closed under sum and scalar multiplcation, hence it is a linear subspace.

To find the dimension, note that all the entries of a lower-triangular matrix over the diagonal must be equal to zero. However, each entry of the matrix under the diagonal and in the diagonal might be any element of $\mathbb{F}$ , any entry that can be choosen add up to the dimension of L, we n such elemnts for the first column, (n-1) for the second column, (n-2) for the third column etc.... Therefore,