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 $p,\;q$.

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 $a,\;b,\;c,\;d$.

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

Identifying some of the basic properties (intercepts, asymptotes, quadrants) of a rational function (quadratic over linear)

This exercise will help you solve equations of type ax-b = c.

This exercise will help you solve equations of type ax-b = c.

Factorise $x^2+cx+d$ into 2 distinct linear factors and then find $\displaystyle \int \frac{ax+b}{x^2+cx+d}\;dx,\;a \neq 0$ using partial fractions or otherwise.

Factorise $x^2+bx+c$ into 2 distinct linear factors and then find $\displaystyle \int \frac{a}{x^2+bx+c }\;dx$ using partial fractions or otherwise.

Displacement-time graphs are given and the student should select the correct velocity-time graphs from a list. Includes linear, piecewise linear and quadratic displacement-time functions.

This exercise will help you solve equations of type ax-b = c.

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

Putting a pair of linear equations into matrix notation and then solving by finding the inverse of the coefficient matrix. 

Solve a system of linear equations using Gaussian elimination.

Three questions on linear combinations and products of matrices.

Solve a pair of linear equations by writing an equivalent matrix equation.

Apply the factor theorem to check which of a list of linear polynomials are factors of another polynomial.

Looking at gradients and values for x and y for straight-line graphs

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 $\phi$.

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 $\phi$.

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

No description given

Putting a pair of linear equations into matrix notation and then solving by finding the inverse of the coefficient matrix. 

A graph is drawn. A student is to identify the derivative of this graph from four other graphs. There are four version of this question: I: cubic, II: linear, III: quadratic, IV: sinusoisal.

Putting a pair of linear equations into matrix notation and then solving by finding the inverse of the coefficient matrix. 

No description given

No description given

No description given