# Few Urgent Linear Algebra Questions

• Nov 16th 2008, 10:54 PM
My Little Pony
Few Urgent Linear Algebra Questions
1. Determine which of the following formulas hold for all invertible nxn matrices A and B

a. A+B is invertible
b. (I - A)(I + A) = I - A^2
c. A^9 is invertible
d. (A + A^-1)^3 = A^3 + A^-3
e. AB = BA
f. (A+B)^2 = A^2 + B^2 + 2AB

2. Determine the dimensions of the following subspaces of R^2 respectively R^3

1. R^3= span{(-2, -12, 10), (-1, 3, -4), (2, 3, -1)}
2. R^2= span{(–4, –2), (2, 0), (5, 0), (9, –1), (–2, 3)}
3. R^2= span{(7, –2), (–6, 5)}
4. R^2= span{(0, 0)}
5. R^3= span{(–20, –4, 8), (–1, –4, –2), (10, 2, –4), (–4, 3, 4)}
6. R^2= span{(2, –5, –7), (–6, 7, –8), (4, 8, –2)}
• Nov 17th 2008, 12:47 AM
clic-clac
$n \in \mathbb{N}, I$ the $n\times n$ identity matrix, $A$ and $B$ two invertible $n\times n$ matrices.

1)
a) What about $I+(-I)$ ?

b) $(I-A)(I+A)=I^{2}+I.A-A.I-A^{2}=I+A-A-B^{2}=I-A^{2}$.
The first equality is true because $M(n,K)$ is a ring.

c) What about $(A^{9})((A^{-1})^{9})$ and associativity of multiplication.

d) $I^{-1}=I$ , so $(I+I)^{3}=(2I)^{3}=8I\neq I^{3}+I^{-3}=2I$

e)Try with $\begin{pmatrix} 0&1 \\ 1&0 \end{pmatrix}$ and $\begin{pmatrix} 2&0 \\ 0&1 \end{pmatrix}$

f) What we can write in any ring is $(A+B)^{2}=A^{2}+AB+BA+B^{2}$
You get the asked equality iff $AB+BA=2AB$, that is to say iff $AB=BA$. So e) answers the question.

2) You just have to check how many vectors are independant in each set. Note that a subspace of $\mathbb{R}^{n}$ has a maximal dimension of $n$, so if a set of vectors $(x_{i})_{i\in I}$ contains at least $n$ independant vectors, then $span(x_{i}\ ;\ i\in I) = \mathbb{R}^{n}$