What is vector in terms of machine learning

I would think that much of your problem comes because vector is a general term with many uses. In this case, think of it as a list of values or a row in a table. The data structure is a 1-dimensional array; a vector of N elements is an N-dimensional vector, one dimension for each element.

For instance, the input (3.14159, 2.71828, 1.618) is a vector of 3 elements, and could be represented as a point in 3-dimensional space. Your program would declare a 1x3 array (one-dimensional data structure) to hold the three items.

Does this help you visualize the basic input handling? This is not a difficult problem with a Wronkskian transformation matrix -- it's just a change in format and visualization.

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The feature vector is simply one row of input. For instance, in the popular machine learning example of housing price prediction, we might have features (table columns) including a house's year of construction, number of bedrooms, area (m^2), and size of garage (auto capacity). This would give input vectors such as

[1988, 4, 200, 2]
[2001, 3, 220, 1]

etc.

In Simple words,
Dimensions : the attributes/features taken for analysis
eg:
a) In health care domain : height, weight, gender, pulse rate, cholestral level
b) In banking domain : age, gender, profession, marital status etc

n-Dimensional Vector :<e₁, e₂, e₃, ...., e_n> where e_i is the value of dimension i and elements are ordered.
example:
<180, 74, M, 60, 120> is a 6-Dimensional Vector where 180, 74, M, 60, 120 are the values of attributes/dimensions height, weight, gender, pulse_rate, cholesterol_level respectively.

<180, 74, M, 60, 120> and <180, M, 74, 60, 120> are not same as the order of dimensions weight and gender are changed.