305 results authored by Simon Thomas - search across all users.

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• #### Maclaurin and Taylor series

Find the first few terms of the Maclaurin and Taylor series of given functions.

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• #### Simon's copy of Represent a linear map as a matrix with a given basis

Let $P_n$ denote the vector space over the reals of polynomials $p(x)$ of degree $n$  with coefficients in the real numbers.

Let the linear map $\phi: P_4 \rightarrow P_4$ be defined by:

$\phi(p(x))=ap(x) + (bx + c)p'(x) + (x ^ 2 + dx + f)p''(x)$

Using the standard basis for range and domain find the matrix given by $\phi$.

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• #### Simon's copy of Represent a linear map as a matrix given a basis

Let $P_n$ denote the vector space over the reals of polynomials $p(x)$ of degree $n$  with coefficients in the real numbers. Let the linear map $\phi: P_4 \rightarrow P_4$ be defined by: $\phi(p(x))=p(a)+p(bx+c).$Using the standard basis for range and domain find the matrix given by $\phi$.

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• #### Simon's copy of Row reducing a matrix and finding its rank and nullity-MA2223

Reduce a 5x6 matrix to row reduced form and using this find rank and nullity.

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• #### Simon's copy of Are given matrices in (reduced) row echelon form?

Choose which of 5 matrices are in a) row echelon form but not reduced b) reduced row echelon form c) neither.

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• #### Simon's copy of Do three given vectors form a spanning set and are they linearly independent?

Given the following three vectors $\textbf{v}_1,\;\textbf{v}_2,\;\textbf{v}_3$ Find out whether they are a linearly independent set are not. Also if linearly dependent find the relationship $\textbf{v}_{r}=p\textbf{v}_{s}+q\textbf{v}_{t}$ for suitable $r,\;s,\;t$ and integers $p,\;q$.

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• #### Simon's copy of Determine if vectors form a spanning set

Given $5$ vectors in $\mathbb{R^4}$ determine if a spanning set for $\mathbb{R^4}$ or not by looking for any simple dependencies between the vectors.

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• #### Simon's copy of Combine linearly dependent vectors to get zero

Real numbers $a,\;b,\;c$ and $d$ are such that $a+b+c+d=1$ and for the given vectors $\textbf{v}_1,\;\textbf{v}_2,\;\textbf{v}_3,\;\textbf{v}_4$ $a\textbf{v}_1+b\textbf{v}_2+c\textbf{v}_3+d\textbf{v}_4=\textbf{0}$. Find $a,\;b,\;c,\;d$.

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• #### Simon's copy of Invert a 3x3 matrix using row operations

$A$ a $3 \times 3$ matrix. Using row operations on the augmented matrix $\left(A | I_3\right)$ reduce to $\left(I_3 | A^{-1}\right)$.

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• #### Simon's copy of Gauss elimination to solve a system of linear equations.

Solving a system of three linear equations in 3 unknowns using Gauss Elimination in 4 stages. Solutions are all integral.

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