Linear Algebra Explanations on true and false.

Question

1.Could someone prove that if a set of vectors in a $p$-dimensional vector space $Q$ is a spanning set for $Q$, it is a basis.

2.If $T$ is a linear transformation from $\mathbb R^3$ onto $P_2$, then $T$ must be one to one. This is false correct? Since a linear transformation that is onto does not have to be one-to-one.

3.For an inner product space, not necessarily $\mathbb R^n$, there can be vectors of length $0$ but are non zero vectors. Is this true, since length $0$ have something to do with orthogonality?

Any help with these $3$ questions would be greatly appreciated.

Thanks!

Answer 1

1) A basis also has the property of linear independence. So spanning is a necessary, but not sufficient condition for a set of vectors to be a basis.

2) I'm assuming $P_{2}$ is $P_{2}(\mathbb{R})$, the set of polynomials of degree $2$ over the field $\mathbb{R}$? If this is the case, then $P_{2}(\mathbb{R})$ is indeed isomorphic to $\mathbb{R}^{3}$. However, we can define $T: \mathbb{R}^{3} \to P_{2}(\mathbb{R})$ that is neither injective nor surjective. Consider $T(x) = (0, 0, 0)$. This is clearly multiplicative, as $T(kx) = kT(x) = k(0, 0, 0) = 0$. Additivity holds in the same manner. Note that if $T$ is onto, it must be one-to-one, as the dimensions are equal.

3) If you have a vector of zero length, it should also be a zero vector. The consequence of a zero length can be seen in $\mathbb{C}$, as an example. We have that $u \cdot v = \overline{v \cdot w}$, rather than $u \cdot v = v \cdot u$.

Answer 2

1) Not true. To be a basis it also has to be linearly independent.

2) What do you mean by $P_2$? If it has dimension $3$, then a linear transformation from ${\mathbb R}^3$ onto it must be one-to-one. If the dimension is greater than $3$, there are no linear transformations from ${\mathbb R}^3$ onto it. If the dimension is less than $3$, there are no one-to-one linear transformations from ${\mathbb R}^3$ onto (or into) it.

3) Part of the definition of inner product space is that any vector other than $0$ has nonzero length.