Solving Linear EquationsVectors and Linear Equations The row picture shows tow lines meeting at a single point(solution). The column picture combines the column vectors on the left side to product th...
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Introduction to VectorsVectors and Linear CombinationsVECTOR ADDITION $v =\begin{bmatrix}v_1 \v_2 \\end{bmatrix}$ and $w =\begin{bmatrix}w_1 \w_2 \\end{bmatrix}$ add to $v + w = \begin{bmatrix}v_1 + w...
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Vector Spaces and SubspacesSpaces of VectorsThe space $R^n$ consists of all column vectors $v$ with n components. “Inside the vector space” means that the result stays in the space. A subspace of a ve...
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OrthogonalityOrthogonality of the Four SubspacesTwo vectors are orthogonal when their dot product is zero: $v \cdot w = 0$ or $v^Tw = o$. ProjectionsWhen $Ax=b$ has no solution, multiply by $A^T$ an...
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Linear TransformationsThe Idea of a Linear TransformationA transformation $T$ assigns an output of $T(v)$ to each input vector $v$ in $V$. The transformation is linear if it meets these requirements f...
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DeterminantsThe determinant is zero when the matrix has no inverse. $$\begin{vmatrix}a & b \ c & d \end{vmatrix}=ad-bc$$ The Properties of Determinants The determinant of the $n$ by $n$ ident...
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Eigenvalues and EigenvectorsIntroduction to EigenvaluesThe basic equation is $Ax = λx$. The number $λ$ is an eigenvalue of $A$.The vector $x$ is an eigenvector of $A$. The Equation for the Eigenvalues...
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