Basis and dimension

Japanese school year: University year 1

What you learn

Learn the concepts of a basis, which spans a vector space with linearly independent vectors, and dimension, representing the number of basis vectors. These are essential for understanding degrees of freedom in vector spaces. Familiarity with linear combinations and linear independence is helpful.

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Key points

This formula indicates that vectors $\boldsymbol{v}_1, \dots, \boldsymbol{v}_n$ span the entire vector space $V$ . Any vector in $V$ can be formed simply by multiplying these vectors by numbers and adding them together.

V = \operatorname{span}(\boldsymbol{v}_1, \dots, \boldsymbol{v}_n)

This formula calculates the dimension of the sum of two subspaces, $W_1 + W_2$ . Add the dimensions of $W_1$ and $W_2$ , then subtract the dimension of their overlapping intersection $W_1 \cap W_2$ .

\dim(W_1 + W_2) = \dim(W_1) + \dim(W_2) - \dim(W_1 \cap W_2)

This property shows that the dimension of a subspace $W$ inside space $V$ can never exceed the dimension of $V$ . If their dimensions are equal, $W$ and $V$ are the exact same space.

\dim(W) \le \dim(V) \quad (W \subseteq V)

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Practice 1Coming soon