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2021-01-11 What is the vector equation for a straight line?? $$ \pmb{r} =\pmb{a} + \lambda \pmb{b} $$ $$\pmb{r} =\pmb{a} + \lambda \pmb{b}$$ What is the variable $\pmb{a}$ called?? The position vector. $$\pmb{r} =\pmb{a} + \lambda \pmb{b}$$ What is the variable $\pmb{b}$ called?? The direction vector. $$\pmb{r} =\pmb{a} + \lambda \pmb{b}$$ What is the variable $\pmb{r}$ called?? The general position vector for a point on the line. $$\pmb{r} =\pmb{a} + \lambda \pmb{b}$$ What is the informal name for $\lambda$ called here?

See also: [[Further Maths - Vector Equation of a Line]]S [[Further Maths - Vector Equation of a Plane]]S [[Further Maths - Dot Product]]S What is the formula for the distance to a three dimensional point $(a,b,c)$?? $$ \sqrt{a^2 + b^2 + c^2} $$ $$ \left(\begin{matrix} a \ b \ c \end{matrix}\right) $$ What is the formula for the length of the vector?? $$ \sqrt{a^2 + b^2 + c^2} $$

What is a linear transformation?? A transformation that only involves linear terms in $x$ and $y$. $$\left(\begin{matrix}x \ y\end{matrix}\right) \mapsto \left(\begin{matrix}x + y \ y - 1\end{matrix}\right)$$ Is that a linear transformation?? No because it’s a translation. $$\left(\begin{matrix}x \ y\end{matrix}\right) \mapsto \left(\begin{matrix}2x - y \ x + y\end{matrix}\right)$$ Is that a linear transformation?? Yes. $$\left(\begin{matrix}x \ y\end{matrix}\right) \mapsto \left(\begin{matrix}2y \ -x^2\end{matrix}\right)$$ Is that a linear transformation?? No because it involves $x^2$ terms. If something is a linear transformation of $$\left(\begin{matrix}x \ y\end{matrix}\right)$$, how could you write the resulting matrix?

Other than matricies, how can you solve a system of three equations?? Using the triangle method. How does the triangle method of solving a system of three equations work?? Using back substitution. What are the steps involved in the triangle method?? Add/subtract two equations of three variables to form an equation of two variables Do this for another equation, forming an equation with the same two variables Solve this using simulataneous equations Substitute the new values and solve for the final variable Why is it called the “triangle” method of solving equations?

What is the crucial idea that allows you to solve systems of equations using matricies?? If $$ A\left( \begin{matrix} x \\ y \ z\end{matrix} \right) = v $$ then $$ \left( \begin{matrix} x \\ y \ z\end{matrix} \right) = A^{-1}v $$ How could you rewrite [2x+3y = 4 \\ 4x-y = 7] as the product of two matricies?? $$ \left(\begin{matrix}2 & 3 \\ 4 & -1\end{matrix}\right)\left(\begin{matrix}x \ y\end{matrix}\right) = \left(\begin{matrix}4 \ 7\end{matrix}\right) $$ How could you solve this $$\left(\begin{matrix}2 & 3 \\ 4 & -1\end{matrix}\right)\left(\begin{matrix}x \ y\end{matrix}\right) = \left(\begin{matrix}4 \ 7\end{matrix}\right)$$?

What is the math definition of the inverse matrix of $M$?? The inverse matrix of $M$ is the matrix $M^{-1}$ such that $MM^{-1} = M^{-1}M = I$. What’s the wordy definition for an inverse matrix of $M$?? The matrix that when multiplied by the original matrix yields the identity matrix. What’s the formula for the inverse of matrix $$M = \left( \begin{matrix}a & b \\ c & d\end{matrix} \right)$$?? $$ \frac{1}{\text{ad - bc} M} \left( \begin{matrix}d & -b \\ -c & a\end{matrix} \right) $$

What is the determinant of a matrix?? A special value associated with a matrix. What does $|M|$ mean in the context of matricies?? The determinant of the matrix $M$. What does $\text{det} M$ mean?? The determinant of matrix $M$. What’s the formula for the determinant of a $2x2$ matrix?? $$ ad - bc $$ What is $M$ if $\text{det} M$ is $0$?? $M$ is a singular matrix. What is $M$ if $\text{det} M$ is not $0$?

See also: [[Further Maths - Determinants]]S [[Further Maths - Inverting Matricies]]S [[Further Maths - Solving Systems of Equations Using Matricies]]S [[Further Maths - Linear Transformations]]S If a matrix is $m \times n$, how many columns does it have?? $m$ columns. If a matrix is $m \times n$, how many rows does it have?? $n$ rows. What are the dimensions of a matrix?? $$ \text{rows} \times \text{columns} $$