# Linear Algebra - Basis & Dimension ## Basis ### Definition A basis for vector space is a sequence of vector that is linearly independent and that spans the space. ### Examples 1. basis of $\mathbb{R}^n=\{\left[ {\begin{array}{} 1\\0\\0\\\vdots\\0\end{array} } \right],\left[ {\begin{array}{} 0\\1\\0\\\vdots\\0\end{array} } \right],...,\left[ {\begin{array}{} 0\\0\\0\\\vdots\\1\end{array} } \right]\}$ 2. basis of $V=\{\left[ {\begin{array}{} x\\y\\z\end{array} } \right]\mid x+y+z=0\}$ is $\{\left[ {\begin{array}{} -1\\1\\0\end{array} } \right],\left[ {\begin{array}{} -1\\0\\1\end{array} } \right]\}$ ## Dimension ### Definition The dimention of a vector space is the number of vectors in any of its bases. ## Commen Question ### Row(A)'s Basis & Dimension :::info Find basis of Row(A) ::: :::spoiler Find R (the reduce echelon form of A), the row with **leading variables** of **R** is the basis of Row(A) ::: :::info Find dimension of Row(A) ::: :::spoiler Find R (the reduce echelon form of A), and count the **leading variables**. ::: ### Col(A)'s Basis & Dimension :::info Find basis of Col(A) ::: :::spoiler Find R (the reduce echelon form of A), the row with **leading variables** of **A** is the basis of Row(A) ::: :::info Find dimension of Col(A) ::: :::spoiler Find R (the reduce echelon form of A), and count the **leading variables**. ::: ### ker(A)'s Basis & Dimension :::info Find basis of ker(A) ::: :::spoiler Find the solution of $A\vec{x}=\vec{0}\rightarrow S={u_1,u_2,...u_d}$ Since $Span(S)=ker(A)$ and $c_1u_1+c_2u_2+...+c_du_d=\vec{0}$ (S is linear independent), S is basis of ker(A) ::: :::info How to find dimension of ker(A) ::: :::spoiler Find R (the reduce echelon form of A), and count the **free variables**. :::