5 Vector Spaces Free Online FlashCards

What is an abstract vector space?

A vector space is a set whose objects can be added and multiplied by scalars while satisfying the usual vector-space rules.

What is the subspace test?

A nonempty subset is a subspace when it is closed under addition and scalar multiplication; equivalently, it is closed under all linear combinations.

What is a linear combination?

A linear combination has the form $c_1v_1+\cdots+c_kv_k$, where the $c_i$ are scalars.

What is $\operatorname{span}(S)$?

The span of $S$ is the set of every linear combination of vectors in $S$. It is always a subspace.

When are vectors linearly independent?

A set is linearly independent when $c_1v_1+\cdots+c_kv_k=0$ has only the solution $c_1=\cdots=c_k=0$.

How can row reduction test column independence?

Place the vectors as matrix columns. They are independent exactly when the homogeneous system has only the zero solution, equivalently when every column contains a pivot after row reduction.

What two properties define a basis?

A basis is a linearly independent spanning set. Thus it contains no redundant vector and can represent every vector in the space.

What does a basis guarantee about representation?

Every vector has exactly one representation as a linear combination of the vectors in a given basis.

What is the dimension of a finite-dimensional space?

The dimension is the number of vectors in any basis. All bases of a finite-dimensional vector space have the same number of vectors.

What is $\dim(P_n)$?

The dimension of $P_n$, the polynomials of degree at most $n$, is $n+1$, using the basis $\{1,x,\ldots,x^n\}$.

What are coordinates relative to an ordered basis?

For an ordered basis $\mathcal{B}=\{v_1,\ldots,v_n\}$, $[v]_{\mathcal{B}}$ is the column of coefficients in $v=c_1v_1+\cdots+c_nv_n$.

How does the basis matrix recover coordinates?

If $P_{\mathcal{B}}=[v_1\ v_2\ \cdots\ v_n]$, then $v=P_{\mathcal{B}}[v]_{\mathcal{B}}$. For a basis in $\mathbb{R}^n$, $[v]_{\mathcal{B}}=P_{\mathcal{B}}^{-1}v$.