I am unsure of my answers here any confirmation on my answers, additional knowledge or input would be awesome.
a) All vectors of the form $(a,b,c)$ where $a+b+c = 0$ form a subspace in $R^3$ True or False? why?
I believe this is true since this would mean it contains the zero vector which is required for a subspace, but I am unsure is this right?
b) The set of all 3 by 3 matrices is a vector space of dimension 6? True or False? Why?
False, because this forms a system of equations in only 3 dimensions?
c) The set of all polynomials of degree 3 is a vector space. True or False?Why?
No clue on this one.
d) Any four vectors in $R^5$ are linearly dependent. True or False? Why?
True, because depending on what our 5th vector is it could change the whole problem to be linearly independent.
Best Answer
a) true
just use the axioms. suppose $(a,b,c)$ and $(d,e,f)$ satisfy your condition. $(a,b,c)+x*(d,e,f) = (a+xd,b+xe,c+xf)$ => $a+xd+b+xe+c+xf = (a+b+c)+x*(d+e+f) = 0$
b) false
it's a 9-dimensional vector space
c) false
because: $(x^3 + 1)$ is a degree 3 polynomial. $(-x^3 + x)$ is a degree 3 polynomial. but $(x^3+x) + (-x^3 + x) = (x+1)$ is not a degree 3 polynomial but a degree 1 polynomial.
d) false
the five unit vectors are independent. so, the first 4 of them are independent too. so, not every 4 vectors are dependent.