If A is a 3 by 3 matrix which gives a rotation about some line through the origin in R^3 , then columns of A form a basis of R^3
For any matrix A, the image of A^7 is contained in the image of A
Every inner product space has an orthonormal basis.
Above are true or false statements.
Can anyone help me with whether these are true or false?
I have no idea about 1 and 2
For 3, every finite inner product space has an orthonormal basis by gram-schimith process,
does it hold for infinite space?
Some Hints:
If the columns of the $3 \times 3$ matrix $A$ do not form a basis of $\mathbb{R}^3$, then the columns are linearly dependent, and hence, $A$ has a non-trivial nullspace, i.e. there is a non-zero vector $x$ such that $Ax = 0$. Is this possible if $A$ is a rotation matrix?
For any vector $x$ in the image of $A^7$, we have $x = A^7y$ for some vector $y$. To show that $x$ is also in the image of $A$, we need to write $x = Az$ for some vector $z$. Can you see how to do this?
See this question.