Linear algebra (線形代数) is the branch of mathematics that uses vectors and matrices to handle many quantities linked by linear expressions all at once.
A linear expression, such as y = 2x + 3, is one built only by adding terms of the form constant times variable and a constant. Solving a system of linear equations in one go, or transforming many coordinates at once as when an image is rotated or enlarged, can be described with the ideas of linear algebra. It also underpins computer graphics and machine learning, which process large volumes of numbers in table form.
AnalogyIt is like a building kit with fixed parts (sets and tables of numbers) and fixed assembly steps (multiplication and addition): the same steps process a large batch of numbers together.
How to remember"Linear" comes from "line": the graph of a linear expression is a straight line.
A vector (ベクトル) is a quantity that has both magnitude and direction, and it can be written as a set of numbers arranged in a single line.
For example, a move of 4 to the right and 3 up on a plane is the vector (4, 3): the arrow's length, 5, is its magnitude, and the way the arrow points is its direction. A quantity that has magnitude only is called a scalar (スカラー), with temperature and mass as typical examples.
AnalogyDirections such as "300 m northeast of the station" give both a distance and a bearing, so they can be drawn as a single arrow.
How to remember"Vector" comes from the Latin for "carrier": picture an arrow that carries a point from one place to another.
Exam points
Mix-upA vector (ベクトル) has magnitude and direction, and what separates it from a scalar (スカラー), which has magnitude only, is whether it has a direction.
A matrix (行列) is a set of numbers laid out in a rectangle made of horizontal lines (rows) and vertical lines (columns).
Row \ Column
Column 1
Column 2
Column 3
Row 1
2
0
7
Row 2
1
5
3
A matrix with m rows and n columns is called an m-by-n matrix and holds m × n numbers, its elements (成分). A matrix with only one row or only one column can be treated as a vector, so a matrix can also be seen as vectors placed side by side. Matrices are used to write the coefficients of a system of linear equations compactly and to carry out a coordinate transformation in one calculation.
AnalogyThink of a cinema seating chart: the row and seat number fix each position, and every position holds exactly one number.
How to rememberRows run across like a line of text, while columns stand upright like the columns holding up a building.
Exam points
Mix-upIn a matrix (行列), a row (行) runs horizontally and a column (列) runs vertically, so a 2-by-3 matrix has two stacked tiers, each holding 3 numbers side by side.
Vector vs. Matrix
Aspect
Vector
Matrix
Key differenceHow numbers are arranged
A set of numbers in a single line
A table laid out as a rectangle of rows and columns
How its shape is described
The number of components (two means an arrow on a plane)
The number of rows and columns (m by n)
How they relate
Can be seen as a matrix with only one row or one column
In differentiation and integration (微分と積分), differentiation finds the rate of change at a point (the slope of the graph), while integration adds up finely divided pieces to find a total amount (such as an area).
Differentiation
Integration
What it finds
Rate of change at a point
Accumulated total
On a graph
Slope of the tangent line
Area between the graph and the horizontal axis
For example, differentiating position with respect to time gives the speed at each instant, and integrating speed over time gives the distance travelled. The branch of mathematics that deals with such changing quantities is called analysis (解析).
AnalogyThe flow rate from a tap corresponds to what differentiation gives, and the amount of water that builds up in the bucket corresponds to what integration gives.
How to rememberDifferentiation looks at the tiny difference between nearby values, while integration, like "integrate" in everyday English, puts the pieces together into a whole.
Exam points
Mix-upDifferentiation (微分) and integration (積分) are inverse operations: integrating the rate of change obtained by differentiating a quantity tells you how much the original quantity changed.
An error (誤差) is the difference (the gap) between a value obtained by measurement or calculation and the true value.
Type
When it occurs
Example
Rounding error
Trimming digits to fit a set number of places
Writing 1/3 as 0.333 leaves a gap of about 0.00033
Absorption
Adding or subtracting numbers of very different sizes
With 7 significant digits, adding 0.0001 to 10000000 leaves no trace of the 0.0001
Because measuring instruments have scales of limited fineness and computers can hold numbers only to a limited number of digits, error usually cannot be brought down to zero. In computer calculations, for instance, rounding error (丸め誤差) arises when a value is trimmed to fit a set number of digits, and absorption (情報落ち) occurs when numbers of very different magnitudes are added or subtracted and the smaller one fails to show up in the result.
AnalogyWhen you measure a desk with a tape measure, anything between two marks has to be judged by eye, so however carefully you measure, a slight gap from the true width remains.
How to remember"Error" comes from the Latin errare, "to wander": picture measured values wandering around the true value.
Exam points
ExamA value made by cutting off an irrational number or a non-terminating decimal at a finite number of digits is an approximate value (近似値), and it contains a difference from the true value (真の値), that is, an error (誤差).