for a given vector, find its coordinates with respect to a specific basis


obtain the dot product of two vectors

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Context Panel: Dot Product (apply to sequence of two vectors)

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Example 12.1: Obtain the dot product of two vectors using the Common Symbols palette or using a period


determine the angle between two vectors

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Use the VectorAngle command in the Student LinearAlgebra package


calculate a vector norm

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Example 12.2: Calculate the norm of a vector using symbols or a command


project one vector onto another


project a vector onto a subspace spanned by two other vectors or onto a plane through the origin


obtain $\mathbf{A}\times \mathbf{B}$, the crossproduct of two vectors

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Math mode: use $\times$ from Common Symbols, or Operators palettes
Text mode: use &x
See Example 12.3


extract a maximal linearly independent subset from a set of vectors

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Use the Basis command from the Student LinearAlgebra package


obtain the determinant of a matrix

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Context Panel: Standard Operations → Determinant

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Example 12.4: Use the absolute value template from the Layout palette

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Use the Determinant command from the Student LinearAlgebra package


multiply a matrix by a scalar

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In math mode, use a space as the multiplication operator

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In text mode, use * as the multiplication operator


apply the function $f$ to each element of a vector or matrix $A$

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Use the elementwise operator: $f~\left(A\right)$


obtain the product of two matrices $A$ and $B$

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Use the period for noncommutative multiplication: $A\.B$


raise a square matrix $A$ to a positive integer power such as 3

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Use ordinary exponentiation: ${A}^{3}$


obtain the rank of a matrix

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Context Panel: Queries → Rank

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Use the Rank command from the Student LinearAlgebra package


obtain the nullity of a matrix


obtain bases for row, column, and null spaces of a matrix


obtain the transpose or Hermitian transpose of a matrix

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Context Panel: Standard Operations → Transpose

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Example 12.5: In Math mode, for a matrix A, its transpose can be found by typing ${A}^{\mathrm{\%T}}$


construct a projection matrix


perform augmentation or stacking operations on a matrix

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Example 12.6: Stacking $A$ on top of $B$, where $A$ and $B$ are vectors or matrices is done by typing $\u27e8A\,B\u27e9$; Augmenting $A$ with $B$ is done by typing $\u27e8A\B\u27e9$


solve the linear system $A\mathbf{x}\=\mathbf{b}$

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Augment by using $\u27e8A\b\u27e9$ and apply Context Panel: Solvers and Forms → RowEchelon Form (see Example 12.6)

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Use the LinearSolve command from the Student LinearAlgebra package


implement Gaussian elimination

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Context Panel: Solvers and Forms → RowEchelon Form


obtain the inverse of a square matrix $A$

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In math mode, simply execute ${A}^{1}$

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In text mode, execute A^(1)

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Context Panel: Standard Operations → Inverse

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Use the MatrixInverse command from the Student LinearAlgebra package


obtain the pseudoinverse of a singular or nonsquare matrix

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Context Panel: Standard Operations → Pseudoinverse

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Use the Pseudoinverse command from the Student LinearAlgebra package


obtain eigenvalues and eigenvectors for a matrix

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Context Panel: Eigenvalues, etc → Eigenvalues


compute ${e}^{At}$ for a constant matrix $A$


apply the GramSchmidt process to the columns of a matrix, or a list or set vectors


apply the GramSchmidt process to a list or set of vectors

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Use the GramSchmidt command from the Student LinearAlgebra package


visualize the effect of multiplying a planar vector by a square matrix


Equate corresponding components in two vectors or matrices

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Context Panel: Equate (applied to the sequence of objects)


convert linear equations to matrix form

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Load Student[LinearAlgebra]

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Context Panel: Student Linear Algebra → Constructions → Generate Matrix (applied to sequence of equations)

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Use the GenerateMatrix command from the Student LinearAlgebra package

